| import OAI.Analysis.FiniteFactor.OrbitSampling.SecondMarginal | |
| namespace OAI | |
| namespace FiniteFactor | |
| open MeasureTheory Filter Finset | |
| open scoped Classical ENNReal | |
| noncomputable section | |
| def sourceSamplingRectangle (E G : SourceCoordinate → Set ℝ) : | |
| Set ((SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := | |
| {p | ∀ i, p.1 i ∈ E i ∧ p.2 i ∈ G i} | |
| theorem sourceSamplingRectangle_measurable (E G : SourceCoordinate → Set ℝ) | |
| (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) : | |
| MeasurableSet (sourceSamplingRectangle E G) := by | |
| simp only [sourceSamplingRectangle,Set.ofPred_forall] | |
| apply MeasurableSet.iInter | |
| intro i | |
| exact ((hE i).preimage ((measurable_pi_apply i).comp measurable_fst)).inter | |
| ((hG i).preimage ((measurable_pi_apply i).comp measurable_snd)) | |
| theorem sourceSamplingProduct_first_indicator (I : Finset SourceCoordinate) (stage : ℕ) | |
| (E G : SourceCoordinate → Set ℝ) | |
| (houtside : ∀ i ∉ sourceSamplingPrefix stage, | |
| Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i) | |
| (θ x : SourceCoordinate → ℝ) (n : ℤ) | |
| (hx : ∀ i, x i ∈ Set.Ico (0 : ℝ) 1) : | |
| sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x = | |
| if (x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G | |
| then sourceSamplingWeight I stage θ n x else 0 := by | |
| have hy (i) : coordinateRotate θ n x i ∈ Set.Ico (0 : ℝ) 1 := | |
| ⟨Int.fract_nonneg _,Int.fract_lt_one _⟩ | |
| have he : (∀ i ∈ sourceSamplingPrefix stage, x i ∈ E i ∧ coordinateRotate θ n x i ∈ G i) ↔ | |
| (x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G := by | |
| constructor | |
| · intro h i | |
| by_cases hi : i ∈ sourceSamplingPrefix stage | |
| · exact h i hi | |
| · exact ⟨(houtside i hi).1 (hx i),(houtside i hi).2 (hy i)⟩ | |
| · intro h i _ | |
| exact h i | |
| unfold sourceSamplingProduct samplingOrientedWeight samplingTestedFirstWeight | |
| simp only [Bool.false_eq_true,ite_false,← coordinateRotate_phase] | |
| simp only [Finset.prod_ite_zero,he] | |
| rfl | |
| /-- Inverse rotation exchanges the two actual tested product orientations. -/ | |
| theorem sourceSamplingProduct_second_inverse (I : Finset SourceCoordinate) (stage : ℕ) | |
| (E G : SourceCoordinate → Set ℝ) (θ y : SourceCoordinate → ℝ) (n : ℤ) | |
| (hy : ∀ i, y i ∈ Set.Ico (0 : ℝ) 1) : | |
| sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ | |
| (coordinateRotate θ (-n) y) = | |
| sourceSamplingProduct stage true (fun i => decide (i ∈ I)) E G n θ y := by | |
| have hinv : coordinateRotate θ n (coordinateRotate θ (-n) y) = y := by | |
| simpa only [neg_neg] using coordinateRotate_neg_cube θ y (-n) hy | |
| apply Finset.prod_congr rfl | |
| intro i _ | |
| have hi := congrFun hinv i | |
| rw [coordinateRotate_phase] at hi | |
| simp only [samplingOrientedWeight,Bool.false_eq_true,ite_false,ite_true,samplingTestedFirstWeight,hi] | |
| rw [coordinateRotate_neg_phase] | |
| /-- The restricted orbit-pair summand is exactly the pushforward of its | |
| first tested product, with no assumed marginal law. -/ | |
| theorem sourceSamplingSummand_restrict_rectangle (I : Finset SourceCoordinate) (stage : ℕ) | |
| (E G : SourceCoordinate → Set ℝ) | |
| (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) | |
| (houtside : ∀ i ∉ sourceSamplingPrefix stage, | |
| Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i) | |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : | |
| (sourceSamplingSummand I stage θ n).restrict (sourceSamplingRectangle E G) = | |
| (sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal | |
| (sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x))).map | |
| (fun x => (x,coordinateRotate θ n x)) := by | |
| have hp : Measurable (fun x : SourceCoordinate → ℝ => (x,coordinateRotate θ n x)) := | |
| measurable_id.prodMk (coordinateRotate_preserving θ n).measurable | |
| have hR := sourceSamplingRectangle_measurable E G hE hG | |
| unfold sourceSamplingSummand sourceSamplingWeightedMeasure | |
| rw [Measure.restrict_map hp hR,restrict_withDensity (hp hR),← withDensity_indicator (hp hR)] | |
| congr 1 | |
| apply withDensity_congr_ae | |
| filter_upwards [sourceSamplingHaar_cube] with x hx | |
| rw [sourceSamplingProduct_first_indicator I stage E G houtside θ x n hx] | |
| change (if (x,coordinateRotate θ n x) ∈ sourceSamplingRectangle E G | |
| then ENNReal.ofReal (sourceSamplingWeight I stage θ n x) else 0) = _ | |
| split_ifs <;> simp | |
| end | |
| end FiniteFactor | |
| end OAI | |